The magnetic moment of a current-carrying circular loop is given by the formula $ M = I \times A $, where $ I $ is the current and $ A $ is the area of the loop.

Assuming the current ($ I $) is the same for both coils, the magnetic moment is directly proportional to the area of the coil ($ M \propto A $).

For two coils with radii $ r_1 $ and $ r_2 $, the areas are $ A_1 = \pi r_1^2 $ and $ A_2 = \pi r_2^2 $.

Given that the ratio of the radii is $ 1:2 $, we can write:

$ \frac{M_1}{M_2} = \frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} $

Thus, the ratio of their respective magnetic moments is $ 1:4 $.